On embeddings between spaces of functions of generalized bounded variation

TL;DR

This paper establishes new embedding relations between various function spaces of generalized bounded variation, providing sufficient conditions and extending known results in Fourier series theory.

Contribution

It introduces novel embedding conditions between function spaces of generalized bounded variation, including new results even for classical spaces.

Findings

Derived sufficient conditions for embeddings between function spaces.

Extended known relationships between classical variation spaces.

Provided new inclusions for spaces related to Fourier series.

Abstract

In this note, we aim to establish a number of embeddings between various function spaces that are frequently considered in the theory of Fourier series. More specifically, we give sufficient conditions for the embeddings $\Phi V[h]\subseteq \Lambda\text{BV}^{(p_n\uparrow p)}$, $\Lambda V[h_1]^{(p)}\subseteq\Gamma V[h_2]^{(q)}$ and $\Lambda\text{BV}^{(p_n\uparrow p)}\subseteq\Gamma\text{BV}^{(q_n\uparrow q)}$. Our results are new even for the well-known spaces that have been studied in the literature. In particular, a number of results due to M. Avdispahi\'{c}, that describe relationships between the classes $\Lambda\text{BV}$ and $V[h]$, are derived as special cases.